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arXiv · 2501.15033

Anisotropic quadratic equations in three variables

Jiamin Li·Jianya Liu
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Abstract

Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to $f(x_1, x_2, x_3) = t$, there are infinitely many such solutions for which (i) $x_1$ has at most $6$ prime factors, and (ii) the product $x_1 x_2$ has at most $16$ prime factors. Various methods, such as algebraic theory of quadratic forms, harmonic analysis, Jacquet-Langlands theory, as well as combinatorics, interact here, and the above results come from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number $6$ or $16$ may be reduced to $5$ or $14$, respectively.

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Jiamin Li, Jianya Liu. 2025-04-18. Anisotropic quadratic equations in three variables. https://arxiv.org/abs/2501.15033

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